Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Timo Keller; Michael Stoll

doi:10.5802/crmath.313

Géométrie algébrique

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
[Vérification exacte de la conjecture BSD forte pour certaines variétés abéliennes absolument simples]

Timo Keller ¹ ; Michael Stoll ¹

¹ Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489.

Résumé
Abstract

Soit $X$ un des $28$ quotients d’Atkin–Lehner d’une courbe $X_{0} (N)$ tel que $X$ est de genre $2$ et sa jacobienne $J$ est absolument simple. On démontre que le groupe de Shafarevich–Tate $Ш (J / ℚ)$ est trivial. Ceci vérifie la conjecture BSD forte pour $J$ .

Let $X$ be one of the $28$ Atkin–Lehner quotients of a curve $X_{0} (N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich–Tate group $Ш (J / ℚ)$ is trivial. This verifies the strong BSD conjecture for $J$ .

Reçu le : 2021-06-30
Révisé le : 2021-11-05
Accepté le : 2021-12-09
Publié le : 2022-05-23

DOI : 10.5802/crmath.313

Classification : 11G40, 11-04, 11G10, 11G30, 14G35

Affiliations des auteurs :

Timo Keller ¹ ; Michael Stoll ¹

¹ Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Timo Keller and Michael Stoll},
     title = {Exact verification of the strong {BSD} conjecture for some absolutely simple abelian surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--489},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.313},
     language = {en},
}

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Timo Keller; Michael Stoll. Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489. doi : 10.5802/crmath.313. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.313/

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